A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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6
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133 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
5
votes
0answers
43 views

Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
5
votes
0answers
217 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
5
votes
0answers
193 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
4
votes
0answers
35 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
4
votes
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53 views

Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space ...
4
votes
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67 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
4
votes
0answers
112 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
4
votes
0answers
96 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
4
votes
0answers
124 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
4
votes
0answers
155 views

Does the non-emptiness of the spectrum of an element of a Banach algebra depend on the Axiom of Choice?

One of the most basic results in functional analysis states that the spectrum of any element of a Banach algebra is non-empty. The proof, as most people might have seen, makes use of Liouville's ...
3
votes
0answers
8 views

Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
3
votes
0answers
23 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
3
votes
0answers
80 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
3
votes
0answers
72 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of unital Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for ...
3
votes
0answers
178 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
0answers
69 views

In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
3
votes
0answers
141 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
3
votes
0answers
67 views

Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest ...
3
votes
0answers
110 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
2
votes
0answers
27 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
2
votes
0answers
22 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
2
votes
0answers
14 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
2
votes
0answers
100 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
2
votes
0answers
26 views

equivalence of properties. Is the restriction in (ii) redundant?

I have a question about the claim, which I found in a paper: Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. The following are equivalent: ...
2
votes
0answers
34 views

Maximal ideal space of $L^{\infty}(m)$ separable?

Let $m$ denote the Lebesgue-measure on the unit interval and let $L^{\infty}(m)$ be the set of measurable essentially bounded functions on that interval (i.e. $f \in L^{\infty}(m)$ if and only if ...
2
votes
0answers
53 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
2
votes
0answers
37 views

Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
2
votes
0answers
36 views

positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
2
votes
0answers
13 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
2
votes
0answers
28 views

Banach algebras for which the Gelfand transform is 1-1 but not onto

Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto? Context Page 96 of Kaniuth's ...
2
votes
0answers
17 views

Lifting an element isometrically (or contractively) from a quotient of a Banach algebra

Let $A$ be a (unital) Banach algebra and let $J$ be an ideal. Let $A_r$ be a closed linear subspace of $A$. Suppose that the continuous linear bijection $A_r/(J\cap A_r)\rightarrow (A_r+J)/J$ is an ...
2
votes
0answers
52 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
2
votes
0answers
152 views

Semigroups: Entire Elements (I)

Problem Given a Banach space $E$. Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$ Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined ...
2
votes
0answers
74 views

Riesz Projection in Functional Analysis.

By definition the Riesz projection of a Banach algebra element $a$ associated with a complex number $\alpha$ is given by $p(\alpha , a)= \frac{1}{2\pi i} \int_{\Gamma} ( \mu -a)^{-1} \,\, ...
2
votes
0answers
76 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
2
votes
0answers
40 views

How I can make $\Bbb{C}[x]$ into a Banach algebra?

Let $\Bbb{C}$ the complex field. Define $\Bbb{C}[x]$ as the set of all polynomials with variable $x$. It is known that $\Bbb{C}[x]$ is a algebra. Now the question is this that how I can make it a ...
2
votes
0answers
27 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
2
votes
0answers
66 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
2
votes
0answers
67 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
2
votes
0answers
158 views

Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
2
votes
0answers
79 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
2
votes
0answers
40 views

Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
2
votes
0answers
51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
1
vote
0answers
30 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
1
vote
0answers
18 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
1
vote
0answers
25 views

Show that the ideal generated by an inner function is closed.

Suppose $H^\infty=\{f\in H(U):\exists M<\infty,\,\forall z\in U,\,|f(z)|\le M\}$ is the set of bounded holomorphic functions in the unit disk. It is a Banach algebra with pointwise multiplication. ...
1
vote
0answers
28 views

Ideals of the operator algebra

Let $A$ be a Banach algebra. Is there any relation ship between two-sided closed ideals of $A$ and two-sided closed ideals of the operator algebra $\mathscr B(A)$? Is there any characterization for ...
1
vote
0answers
23 views

Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
1
vote
0answers
32 views

Equivalent statements involving 'little o'

Let $A$ be a Banach algebra and $a\in A$. $\|z(z-a)^{-1}\|=1+o(\frac{1}{z})$ as $z\rightarrow +\infty$ iff $\|(z-a)^{-1}\|^{-1}=z+o(1)$ as $z\rightarrow +\infty$. i.e., $lim_{z\rightarrow ...